L(s) = 1 | + (−0.835 − 2.57i)2-s + (2.27 + 1.65i)3-s + (−4.29 + 3.11i)4-s + (−0.137 + 0.423i)5-s + (2.34 − 7.21i)6-s + (0.809 − 0.587i)7-s + (7.23 + 5.25i)8-s + (1.50 + 4.64i)9-s + 1.20·10-s − 14.8·12-s + (−0.139 − 0.428i)13-s + (−2.18 − 1.58i)14-s + (−1.01 + 0.734i)15-s + (4.18 − 12.8i)16-s + (−1.49 + 4.59i)17-s + (10.6 − 7.75i)18-s + ⋯ |
L(s) = 1 | + (−0.590 − 1.81i)2-s + (1.31 + 0.952i)3-s + (−2.14 + 1.55i)4-s + (−0.0615 + 0.189i)5-s + (0.957 − 2.94i)6-s + (0.305 − 0.222i)7-s + (2.55 + 1.85i)8-s + (0.502 + 1.54i)9-s + 0.380·10-s − 4.30·12-s + (−0.0386 − 0.118i)13-s + (−0.584 − 0.424i)14-s + (−0.260 + 0.189i)15-s + (1.04 − 3.22i)16-s + (−0.362 + 1.11i)17-s + (2.51 − 1.82i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52316 - 0.302912i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52316 - 0.302912i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.835 + 2.57i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-2.27 - 1.65i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (0.137 - 0.423i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (0.139 + 0.428i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.49 - 4.59i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.878 + 0.638i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 4.57T + 23T^{2} \) |
| 29 | \( 1 + (1.60 - 1.16i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.55 - 7.85i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.91 - 4.30i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.43 - 1.04i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + (0.827 + 0.601i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.10 + 3.39i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-11.6 + 8.46i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.52 - 4.68i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 + (1.82 - 5.63i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.34 - 0.974i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 3.43i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.33 + 10.2i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + (1.64 + 5.04i)T + (-78.4 + 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27978976460735466285890238608, −9.401580474708156331732544171083, −8.668684092412867538845018996911, −8.362858895883134978098177778677, −7.19948067220437676205940276117, −5.00594734935092021517065560222, −4.18908636257058571581414158225, −3.38662990713589809931648116990, −2.68895519090731991360609253892, −1.51436698318893673174203371048,
0.894340575284045330360530106090, 2.47007840246440174567259289915, 4.16232467223599842832902985890, 5.22132748647412571248868998375, 6.30411317416083776032893079540, 7.14775356469105089334167456141, 7.61999227538996292817035229968, 8.385774303145038851189867948217, 9.067239466780725852419820479018, 9.435181155723760097874483901798